A Menger-like property of tree-cut width

TL;DR

This paper extends the concept of leanness from tree decompositions to tree-cut decompositions, showing that every graph admits a minimum width tree-cut decomposition with an edge-connectivity property similar to Thomas' leanness.

Contribution

It introduces a Menger-like property for tree-cut width, establishing that such decompositions exist with an edge-connectivity condition analogous to leanness.

Findings

Every graph admits a minimum width tree-cut decomposition with the property.

The property is analogous to Thomas' leanness but for edge-connectivity.

This extends the applicability of leanness concepts to edge-based decompositions.

Abstract

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness [A Menger-like property of tree-width: The finite case. Journal of Combinatorial Theory, Series B, 48(1):67-76, 1990]. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan as a possible edge-version of tree decompositions [The structure of graphs not admitting a fixed immersion. Journal of Combinatorial Theory, Series B, 110:47-66, 2015]. We show that every graph admits a tree-cut decomposition of minimum width that additionally satisfies an edge-connectivity condition analogous to Thomas' leanness.